ERGODIC PROBLEM FOR THE HAMILTON-JACOBI-BELLMAN EQUATION .1. EXISTENCE OF THE ERGODIC ATTRACTOR

The problem of the convergence of the terms lambda u(lambda)(x), not e qual u(x, T) in the Hamilton-Jacobi-Bellman equations (HJBs) as lambda tends to +0, T goes to +infinity, to the unique number is called the ergodic problem of the HJBs. We show in this paper what kind of qualit ative properties exist behind this kind of convergence, The existence of the ergodic attractor is shown in Theorems 1 and 2, Our solutions o f HJBs satisfy the equations in the viscosity solutions sense.